Time evolution of Wigner functions governed by bipartite Hamiltonian system with kinetic
coupling
Ye-Jun Xu∗ and Qiu-Yu Liu
Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui, 230026, China
Hong-Chun Yuan† and Hong-Yi Fan
Department of Physics, Shanghai Jiao Tong University, Shanghai, 200030, China (Received on 4 December, 2009)
For the bipartite Hamiltonian system with kinetic coupling, we derive time evolution equation of Wigner functions by virtue of the bipartite entangled state representation and entangled Wigner operator, which just indicates that choosing a good representation indeed provides great convenience for us to deal with the dynamics problem.
Keywords: entangled Wigner operator, Wigner function evolution, entangled state representation
1. INTRODUCTION
As is well known, Wigner function, introduced by Wigner[1] in 1932, is a quasi-probability distribution to fully describes the state of a quantum system in phase space. Its partial negativity is indeed a good indication of the highly nonclassical character of the quantum states[2]. Now Wigner function has become a very popular tool not only to study the nonclassical properties of quantum states but also to monitor the decoherence of some quantum states by discussing its time evolution[3, 4]. For a density operator ρof a single-mode system, the Wigner function W(x,p) is defined as (~=1)[5, 6]
W(x,p) =Tr[ρ∆(x,p)]
= 1
2π Z D
x+v 2
ρ
x−
v 2
E
e−ipvdv, (1) where∆(x,p)is the single-mode Wigner operator,|xiis the eigenvector of the coordinate operator obeyingX|xi=x|xi,
|xi=π−1/4 exp
−1
2x 2+√
2xa†−1 2a
†2
|0i, (2) a† and a are the Bose creation and annihilation oper-ators, respectively, satisfying
a,a†
= 1; they are re-lated to coordinate operator X and momentum operatorP, namely, X = a+a†/√2 and P = a−a†/i√2. Us-ing the technique of integration within an ordered prod-uct of operators (IWOP)[7, 8] and the vacuum projector
|0ih0|=: exp −a†a
: ( : : denotes normal ordering), we have obtained the explicitly normal ordering form of∆(x,p) ∆(x,p) ≡ ∆(α,α∗) = (3)
= 1
π:e−
(x−X)2−(p−P)2: =1 π:e−
2(a†−α∗)(a−α)
:,
whereα=√1
2(x+ip). Respectively performing integrations overdxandd pleads to
Z
d p∆(x,p) =√1 π: e
−(x−X)2
: =|xihx| (4)
∗Electronic address:yejunxu@mail.ustc.edu.cn †Electronic address:yuanhch@sjtu.edu.cn
and
Z
dx∆(x,p) =√1 π:e
−(p−P)2
: =|pihp|, (5)
where
|pi=π−1/4 exp
−1
2p 2+i√
2pa†+1 2a
†2
|0i (6)
is the eigenvector of the momentum operator satisfying P|pi = p|pi. Thus for a state |ψi, its Wigner func-tion’s marginal distributionshψ|∆(x,p)|ψiare, respectively,
|hx|ψi|2and|hp|ψi|2, this is just Wigner’s original idea of setting up a function inx−pphase space whose marginal distributions are the probability of finding a particle in coor-dinate space and momentum space.
On the other hand, the motion equation of Wigner func-tion has attracted an ever increasing attenfunc-tion for a given system[11]. For example, in Ref.[12], for the Hamiltonian H1=P2/2m+V(X)of a single particle system, one has de-rived the time evolution of the Wigner function governed by H1,
∂
∂t+ p m
∂ ∂x−
dV(x) dx
∂ ∂p
W(x,p,t)
= ∞
∑
l=1~
2
2l ( −1)l (2l+1)!
d2l+1V(X) dx2l+1
∂
∂p 2l+1
W(x,p,t).
(7) In the classical limit~→0, one derives the Liouville
equa-tion
∂
∂t+ p m
∂ ∂x−
dV(x) dx
∂ ∂p
W(x,p,t) =0. (8) In addition, Wigner function with time evolution of a two-body correlated system has been deduced as well, whose Hamiltonian isH2=2m11P
2 1+
1 2m2P
2
2+V(X1−X2)[13]. Thus an interesting question naturally arise: when the Hamiltonian not only is a two-body correlated system, but also contains kinetic coupling, i.e.,
H= 1
2m1 P12+ 1
2m2
where the potentialV(X1−X2)just depends on the relative distance of the two particles, and the termkP1P2is usually called kinetic coupling term, what is the time evolution of Wigner function governed by H? As far as we know, this problem has not been derived analytically in the literature before. This problem is the task of the present paper. By mentioning two-body correlated case, enlightened by the pa-per of Einstein-Podolsky-Rosen (EPR) in 1935[14] who no-ticed that two particles’ relative coordinateX1−X2and total momentumP1+P2are commutative and can be simultane-ously measured, we naturally think of the common eigen-vector|ηiof the relative coordinate operatorX1−X2and the momentum sum operatorP1+P2, (see Eq.(11) below)[15]. Correspondingly, because in the entangled case only those states simultaneously describing two entangled particles can be endowed with physical meaning, phase space should be understood with regard to|ηi. In the following discussion, we mainly obtain the time evolution of the entangled Wigner function governed by the bipartite Hamiltonian in Eq.(9) by the aid of the entangled state representation|ηi.
2. WIGNER FUNCTION FOR TWO-BODY CORRELATED SYSTEM
To begin with, we briefly review the aspects of the Wigner function for the two-body correlated system. In Ref.[16], the so-called entangled Wigner operator has been successfully
established. Based on it, the corresponding Wigner function can be conveniently derived. This entangled Wigner operator is expressed in the entangled state representation|ηias
∆(σ,γ) =
Z d2η
π3 |σ−ηihσ+η|exp(ηγ∗−η∗γ), (10) whereσ=σ1+iσ2,γ=γ1+iγ2and
|ηi=exp
−1
2|η| 2+ηa†
1−η∗a † 2+a
† 1a
† 2
|00i,η=η1+iη2, (11) obeys the eigenvector equations
(X1−X2)|ηi=
√
2η1|ηi,(P1+P2)|ηi=
√
2η2|ηi (12) withXj=
aj+a†j
/√2 andPj=
aj−a†j
/i√2, i.e.,|ηi
is the common eigenstate of the relative position of two par-ticlesX1−X2and their total momentumP1+P2, and spans a complete and orthogonal space[15]
Z d2η
π2 |ηihη|=1, hη|η′
=πδ η−η′
δ η∗−η′∗
.
(13) Using Eq.(11) and |00ih00|=: exp(−a†1a1−a†2a2): and perfoming the integration in Eq.(10) by virtue of the IWOP technique[7, 8], we obtain the normally ordered form of the Wigner operator∆(σ,γ)
∆(σ,γ) =
Z d2η π3 exp
−1
2|σ−η| 2
+ (σ−η)a†1−(σ−η)∗a†2+a†1a†2
|00i
× h00|exp
−1
2|σ+η|
2+ (σ+η)∗a
1−(σ+η)a2+a1a2
eηγ∗−η∗γ
=
Z d2η π3 : exp
h
−|η|2+γ∗−a†1−a2
η+a†2−γ+a1
η∗i
×exp−|σ|2+σa†1−σ∗a†2+σ∗a1−σa2+a1a2+a†1a † 2−a
†
1a1−a†2a2
:
= 1
π2: exp
h
−|σ|2− |γ|2+γ(a†1+a2) +γ∗(a†2+a1) +σ(a†1−a2) +σ∗(a1−a†2)−2a†1a1−2a†2a2
i
= 1
π2: exp[−(a1+a † 2−γ)(a
†
1+a2−γ∗)−(σ−a1+a†2)(σ∗−a †
1+a2)]:, (14)
where we have used the following integral formula[9, 10]
Z d2z π exp
ζ|z|2+ξz+ηz∗=−1ζexp
−ξηζ
,Re(ζ)<0.
(15) By setting
γ=α+β∗,σ=α−β∗ (16)
with α= √1
2(x1+ip1), β= 1
√
2(x2+ip2) and comparing with Eq.(4), Eq.(14) is rewritten as
∆(σ,γ) = 1 π2: exp
h
−2(a†1−α∗)(a1−α)−2(a†2−β∗)(a1−β)
i
:
=∆(α,α∗)∆(β,β∗), (17)
of∆(σ,γ)overd2γ, i.e.
Z
∆(σ,γ)d2γ=1
π|ηihη||η=σ. (18) For the density matrixρof a given two-body correlated sys-tem, the corresponding Wigner function is
Wρ(σ,γ) =
=Tr[ρ∆(σ,γ)] (19)
=d
2η
π3 hσ+η|ρ|σ−ηiexp(ηγ∗−η∗γ).
3. WIGNER FUNCTION EVOLUTION GOVERNED BYH Based on the above preliminaries, we devote this section to studying the time evolution of the Wigner function for the Hamiltonian in Eq.(9). For the convenient discussion, we first introduce
M=m1+m2, µ= (m1m2)/M, xr=X1−X2, P=P1+P2, Pr=µ2P1−µ1P2, µi=
mi M, (20) whereM,µ, P,xr andPr are total mass, reduced mass, to-tal momentum, relative coordinate, and mass-weight rela-tive momentum, respecrela-tively. It then follows by substituting Eq.(20) into Eq.(9) that
H =
1
2M+kµ1µ2
P2 (21)
+
1 2µ−k
Pr2+k(µ2−µ1)PPr+V(xr).
According to the Heisenberg equation
(∂/∂t)ρ=−i[H,ρ], (22)
we obtain ∂
∂thσ+η|ρ|σ−ηi
=−ihσ+η|
1
2M+kµ1µ2
P2+
1 2µ−k
Pr2 +k(µ2−µ1)PPr+V(xr)]ρ|σ−ηi
+ihσ+η|ρ
1
2M+kµ1µ2
P2+
1 2µ−k
Pr2
+k(µ2−µ1)PPr+V(xr)]|σ−ηi. (23)
In order to simplify Eq.(23), we shall obtain the expression ofhη|Pr. By appealing to the Schmidt decomposition of|ηi in the|pirepresentation[17]
|ηi=e−iη1η2
Z d p
p+
√
2η2
E
1⊗ |−pi2e
−i√2η1p, (24)
where⊗stands for direct product.It follows from Eq.(24) andµ2+µ1=1 that
Pr|ηi=e−iη1η2
Z d phµ2
p+√2η2
+µ1p
i p+
√
2η2
E
1
⊗ |−pi2e−i √
2η1p
=
i ∂
∂√2η1
−
(µ1−µ2)η2
√
2
|ηi. (25)
As a result of Eqs.(12) and (25) we have
hσ+η|
1
2M+kµ1µ2
P2=
1
M+2kµ1µ2
(σ2+η2)2hσ+η|,
hσ+η|
1 2µ−k
Pr2=
1 4µ−
k
2 i
∂ ∂(σ1+η1)
+ (µ1−µ2) (σ2+η2)
2
hσ+η|. (26)
Settingσ+η=τ,σ−η=λand using∂/∂(σ1±η1) = (∂/∂σ1±∂/∂η1)/2 and Eqs.(25) and (26) we rewrite Eq.(23) as
i∂
∂thσ+η|ρ|σ−ηi
=
4
M+8kµ1µ2
σ2η2−k(µ2−µ1)
iσ2
∂ ∂σ1
+iη2 ∂ ∂η1
+4(µ1−µ2)σ2η2
+
1 4µ−
k
2 2i(µ1−µ2)
σ2
∂ ∂σ1
+η2 ∂ ∂η1
− ∂
2 ∂σ1∂η1
+4(µ1−µ2)2σ2η2
+Vh√2(σ1+η1)
i
−Vh√2(σ1−η1)
io
hσ+η|ρ|σ−ηi. (27)
i∂
∂thσ+η|ρ|σ−ηi
=
1 µ+2k
σ2η2−ik(µ2−µ1)
σ2
∂ ∂σ1
+η2 ∂ ∂η1
+A
+
1 4µ−
k
2 2i(µ1−µ2)
σ2
∂ ∂σ1
+η2 ∂ ∂η1
− ∂
2 ∂σ1∂η1
hσ+η|ρ|σ−ηi. (28)
where
A≡Vh√2(σ1+η1)
i
−Vh√2(σ1−η1)
i
=2
∞
∑
s=01 (2s+1)!
∂2s+1V√2σ1
∂√2σ1
2s+1 √
2η1
2s+1
. (29)
Substituting Eq.(28) into
∂
∂tWρ(σ,γ,t) = ∂ ∂t
Z d2η
π3 hσ+η|ρ|σ−ηie
ηγ∗−η∗γ, (30)
and performing the integration by parts we derive
i∂
∂tWρ(σ,γ,t) = Z d2η
π3 e
2i(η2γ1−η1γ2)
1 µ+2k
σ2η2+
1 4µ−
1 2k
×
2i(µ1−µ2)σ2 ∂ ∂σ1−
∂ ∂σ1
∂ ∂η1
+2i(µ1−µ2)η2 ∂ ∂η1
+k(µ2−µ1)
−iσ2
∂ ∂σ1−
iη2 ∂ ∂η1
+A
hσ+η|ρ|σ−ηi
=
1 µ+2k
σ2
−i
2 ∂ ∂γ1
+
1 4µ−
1 2k
×
2i(µ1−µ2)σ2 ∂ ∂σ1−
∂ ∂σ1
(2iγ2) +2i(µ1−µ2)
−i
2 ∂ ∂γ1
(2iγ2)
+k(µ2−µ1)
−iσ2
∂ ∂σ1−
i
−i
2 ∂ ∂γ1
(2iγ2)
+A
× Z d2η
π3 e
2i(η2γ1−η1γ2)hσ+η|ρ|σ−ηi
=
1 µ+2k
σ2
−i
2 ∂ ∂γ1
+
1 4µ−
1 2k
×
2i(µ1−µ2)σ2 ∂ ∂σ1−
∂ ∂σ1
(2iγ2) +2i(µ1−µ2)
−i
2 ∂ ∂γ1
(2iγ2)
+k(µ2−µ1)
−iσ2
∂ ∂σ1−
i
−i
2 ∂ ∂γ1
(2iγ2)
+A
Wρ(σ,γ,t). (31)
Therefore, the equation of motion for the Wigner function is i∂
∂tWρ(σ,γ,t)
=
1 µ+2k
σ2
−i
2 ∂ ∂γ1
+
1 4µ−
k
2 2i(µ1−µ2)
σ2
∂ ∂σ1
+γ2 ∂ ∂γ1
−2iγ2
∂ ∂σ1
−ik(µ2−µ1)
σ2
∂ ∂σ1
+γ2 ∂ ∂γ1
+A
Wρ(σ,γ,t). (32)
∂
∂t+
1
2µ[σ2−(µ1−µ2)γ2] +kσ2
∂
∂γ1
+
1
2µ[γ2−(µ1−µ2)σ2] +kγ2
∂
∂σ1− 2
∂V√2σ1
∂√2σ1
∂
∂√2γ2
Wρ(σ,γ,t)
= ∞
∑
s=12(−1)s (2s+1)!
∂2s+1V√2σ 1
∂√2σ1
2s+1
∂2s+1 ∂√2γ2
2s+1Wρ(σ,γ,t). (33)
To see the physical meaning of Eq.(33) more clearly, from Eq.(16) we find that
γ1= 1
√
2(x1+x2),γ2= 1
√
2(p1−p2), σ1=
1
√
2(x1−x2),σ2= 1
√
2(p1+p2), (34) which lead to
1
2µ[σ2−(µ1−µ2)γ2] = 1
√
2
p
1 m1
+p2 m2
,
1
2µ[γ2−(µ1−µ2)σ2] = 1
√
2
p
1 m1−
p2 m2
. (35)
Eq.(33) is equal to
∂
∂t+
1
√
2
p
1 m1
+p2 m2
+(p1+√p2)k
2
∂
∂γ1
+
1
√
2
p
1 m1−
p2 m2
+(p1−√p2)k
2
∂
∂σ1− 2
∂V√2σ1
∂√2σ1
∂
∂√2γ2
Wρ(σ,γ,t)
= ∞
∑
s=12(−1)s (2s+1)!
∂2s+1V√2σ 1
∂√2σ1
2s+1
∂2s+1 ∂√2γ2
2s+1Wρ(σ,γ,t). (36)
It is seen from Eq.(36) that this equation is expressed by center-of-mass coordinate, relative coordinate and relative momentum. Especially, whenk=0, Eq.(36) becomes
∂
∂t+ 1
√
2
p1 m1
+ p2 m2
∂
∂γ1
+√1
2
p1 m1−
p2 m2
∂
∂σ1− 2
∂V√2σ1
∂√2σ1
∂
∂√2γ2
×Wρ(σ,γ,t)
= ∞
∑
s=12(−1)s (2s+1)!
∂2s+1V√2σ 1
∂√2σ1
2s+1
∂2s+1 ∂√2γ2
2s+1Wρ(σ,γ,t),
(37)
which agrees with that of Ref.[13]. Eq.(37) is formally com-parable with Eq.(7) as expected.
It is instructive to consider the classical limit of this equa-tion. For this purpose, we need to reexpress Eq.(29) as
A=2 ∞
∑
s=0~2s+1
(2s+1)!
∂2s+1V√2σ 1
∂√2σ1
2s+1 √
2η1
~
!2s+1
, (38)
and at the same time change e2i(η2γ1−η1γ2) in Eq.(31) into
e2i(η2γ1−η1γ2)/~, then formally set~=0. Provided the
van-ishes. So we have
∂
∂t+
1
√
2
p
1 m1
+p2 m2
+(p1+√p2)k
2
∂
∂γ1 +
1
√
2
p1 m1−
p2 m2
+(p1−√p2)k
2
∂ ∂σ1
−2
∂V√2σ1
∂√2σ1
∂
∂√2γ2
Wρ(σ,γ,t) =0. (39)
In this sense the terms on the right hand side of Eq.(37) con-stitute the quantum mechanical corrections to the classical Liouville equation. Comparing with Eq.(8), Eq.(39) is called the Liouville-like equation for the two-body correlated sys-tem.
4. CONCLUSIONS
At present, Wigner function is also an important tool to transform the operator equation of motion for the density
op-erator into ac-number equation. However, this equation is rather complicated and brings out the nonlocal nature of the Wigner function. In this present work, we have considered a two-body correlated system with kinetic coupling term in Eq.(9) and successfully derived the time evolution equation of the Wigner function (see Eq.(36)) by virtue of the bipar-tite entangled state representation, which just indicates that choosing a good representation indeed provides great conve-nience for us to deal with the dynamics problem.
Acknowledgements
The author would like to thank the referee for his valuable comments and suggestion. This work was supported by the National Natural Science Foundation of China (Grant Nos. 10874174 and 90203002).
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